Connor-Stevens

Main equations

\[c_m \frac{d V}{dt} = -i_m + \frac{I_e}{A}\] \[\tau_m(V) \frac{d m}{dt} = m_{\infty}(V) - m\] \[\tau_h(V) \frac{d h}{dt} = h_{\infty}(V) - h\] \[\tau_n(V) \frac{d n}{dt} = n_{\infty}(V) - n\] \[\tau_a(V) \frac{d a}{dt} = a_{\infty}(V) - a\] \[\tau_b(V) \frac{d b}{dt} = b_{\infty}(V) - b\] \[i_m =\bar{g}\_L (V - E_L) + \bar{g}\_{Na} m^3 h (V - E\_{Na}) + \bar{g}\_{K} n^4 (V - E\_{K}) + \bar{g}\_{A} a^3 b (V-E_A)\]

alpha-values

\(\alpha_n(V) = \frac{0.02mV^{-1} (V + 45.7mV)}{1 - \exp(-0.1mV^{-1} (V + 45.7mV))}\)

\[\alpha_m(V) = \frac{0.38mV^{-1} (V + 29.7mV)}{1 - \exp(-0.1mV^{-1} (V + 29.7mV))}\] \[\alpha_h(V) = 0.266 \exp(-0.05mV^{-1} (V + 48.0mV))\]

beta-values

\(\beta_n(V) = 0.25 \exp(-0.0125mV^{-1} (V + 55.7mV))\)

\[\beta_m(V) = 15.2 \exp(-0.0556mV^{-1} (V + 54.7mV))\] \[\beta_h(V) = \frac{3.8}{1 + \exp(-0.1mV^{-1} (V + 18mV))}\]

time constants

\(\tau_n(V) = \frac{1.0ms}{\alpha_n(V) + \beta_n(V)}\)

\[\tau_m(V) = \frac{1.0ms}{\alpha_m(V) + \beta_m(V)}\] \[\tau_h(V) = \frac{1.0ms}{\alpha_h(V) + \beta_h(V)}\] \[\tau_a(V) = 0.3632ms + \frac{1.158ms}{1.0 + \exp(0.0497mV^{-1} (V + 55.96mV))}\] \[\tau_b(V) = 1.24ms + \frac{2.678ms}{1.0 + \exp(0.0624mV^{-1} (V + 50.0mV))}\]

asymptotic values

\[a_\infty(V) = \left( \frac{0.0761 * \exp(0.0314mV^{-1} (V + 94.22mV))}{1 + \exp(0.0346mV^{-1} (V + 1.17mV))}) \right)^{1 / 3} ms\] \[b_\infty(V) = \left(\frac{1}{1 + \exp(0.0688mV^{-1} (V + 53.3mV))} \right)^4 ms\] \[n_\infty(V) = \alpha_n(V) \tau_n(V)\] \[m_\infty(V) = \alpha_m(V) \tau_m(V)\] \[h_\infty(V) = \alpha_h(V) \tau_h(V)\]

Suitable initial conditions

\[m(t=0) = 0.010\] \[n(t=0) = 0.156\] \[h(t=0) = 0.966\] \[a(t=0) = 0.540\] \[b(t=0) = 0.289\] \[V(t=0) = -68.0 mV\]

Parameter

\[c_m = 0.1 \frac{\mu F}{mm^2}\] \[\frac{I_e}{A} = 0.35 \frac{\mu A}{mm^2}\] \[\bar{g}_L = 0.003 \frac{mS}{mm^2}\] \[\bar{g}_{Na} = 1.2 \frac{mS}{mm^2}\] \[\bar{g}_{K} = 0.2 \frac{mS}{mm^2}\] \[\bar{g}_{A} = 0.477 \frac{mS}{mm^2}\] \[E_L = -17.0 mV\] \[E_{Na} = 55.0 mV\] \[E_K = -72.0 mV\] \[E_A = -75.0 mV\]

Transient Ca2+ Conductance

\[M_{\infty} = \frac{1}{1+ \exp(-(V+57mV) / 6.2mV)}\] \[H_{\infty} = \frac{1}{1+ \exp(-(V+81mV) / 4mV)}\] \[\tau_M = 0.612 ms + \frac{1ms}{\exp(-(V+132mV)/16.7 mV) + \exp((V+16.8mV)/18.2 mV)}\]

if $V < -80 mV$:

\[\tau_H = 1ms \exp((V+467mV)/66.6mV)\]

else:

\[\tau_H = 28ms + 1ms \exp(-(V+22mV)/10.5mV)\]

Ca2+- depentdent K+ Conducatance

\[c_\infty = \left( \frac{[Ca^{2+}]}{[Ca^{2+}] + 3 \mu M} \right) \frac{1}{1+\exp(-(V+28.3mV)/12.6mV)}\] \[\tau_C = 90.3ms - \frac{75.1ms}{1+\exp((-V+46mV)/22.7mV)}\]

The source code is Open Source and can be found on GitHub.